Fermions coupled with self - dual vortex background on a torus ∗

Abstract: We study fermionic zero modes in the self-dual vortex background on an extra torus in 5+1 dimensions. It is shown that the self-dual vortex arises from the symmetric phase and the asymmetric phase of Higgs field in the context of generalized Abelian Higgs model. Then we analyze the Dirac operator on the extra two-dimensional space and the effective Lagrangian of four-dimensional fermion with the vortex background. Solving the Dirac equation, we obtain the fermionic zero modes in two simple cases. It is shown that the vortex background contributes a phase shift to the fermionic zero modes, and this phase is determined by the topology of the extra dimensions, it can be observed through Aharonov-Bohm effect. For the anti-vortex case, the corresponding zero modes present a singular behavior at the zero point of Higgs field. While for the vacuum one, they are non-singular.